xenharmonic (microtonal wiki) - OTC JI 22 sruti scale

This page examines the omnitetrachordality of the 22-tone Indian 'sruti' scale, a 5-limit JI scale with three step sizes -- one of many possible theoretical tunings for Indian classical music. (This scale may be found in the Scala archive as indian.scl .)

scale step	ratio	cents	name
0	1/1	0.000	Sa
1	256/243	90.225	r1
2	16/15	111.731	r2
3	10/9	182.404	R3
4	9/8	203.910	R4
5	32/27	294.135	g1
6	6/5	315.641	g2
7	5/4	386.314	G3
8	81/64	407.820	G4
9	4/3	498.045	Ma
10	27/20	519.551	m2
11	45/32	590.224	m3
12	729/512	611.730	m4
13	3/2	701.955	Pa
14	128/81	792.180	d1
15	8/5	813.686	d2
16	5/3	884.359	D3
17	27/16	905.865	D4
18	16/9	996.090	n1
19	9/5	1017.596	n2
20	15/8	1088.269	N3
21	243/128	1109.775	N4
22	2/1	1200.000	Sa

A = 256/243 (90.225 cents)
b = 81/80 (21.506 cents)
c = 25/24 (70.672 cents)

9/8 = A+2b+c
4/3 = 3A+4b+2c
2/1 = 7A+10b+5c

interval	ratio	step size
1-2	256/243	A
2-3	81/80	b
3-4	25/24	c
4-5	81/80	b
5-6	256/243	A
6-7	81/80	b
7-8	25/24	c
8-9	81/80	b
9-10	256/243	A
10-11	81/80	b
11-12	25/24	c
12-13	81/80	b
13-14	256/243	A
14-15	256/243	A
15-16	81/80	b
16-17	25/24	c
17-18	81/80	b
18-19	256/243	A
19-20	81/80	b
20-21	25/24	c
21-22	81/80	b
22-1	256/243	A

all modes:

`Abcb AbcbAbcbA AbcbAbcbA`	`AbcbAbcbA bcbA AbcbAbcbA`
`bcbA bcbAbcbAA bcbAbcbAA`
`cbAb cbAbcbAAb cbAbcbAAb`
`bAbc bAbcbAAbc bAbcbAAbc`
`Abcb AbcbAAbcb AbcbAAbcb`		`AbcbAbcbA AbcbAbcbA Abcb`
`bcbA bcbAAbcbA bcbAAbcbA`		`bcbAbcbAA bcbAbcbAA bcbA`
`cbAb cbAAbcbAb cbAAbcbAb`		`cbAbcbAAb cbAbcbAAb cbAb`
`bAbc bAAbcbAbc bAAbcbAbc`		`bAbcbAAbc bAbcbAAbc bAbc`
`Abcb AAbcbAbcb AAbcbAbcb`		`AbcbAAbcb AbcbAAbcb Abcb`
`bcbA AbcbAbcbA AbcbAbcbA`		`bcbAAbcbA bcbAAbcbA bcbA`
		`cbAAbcbAb cbAAbcbAb cbAb`
		`bAAbcbAbc bAAbcbAbc bAbc`
		`AAbcbAbcb AAbcbAbcb Abcb`
	`AbcbAbcbA Abcb AbcbAbcbA`	`AbcbAbcbA AbcbAbcbA bcbA`
	`bcbAbcbAA bcbA bcbAbcbAA`
	`cbAbcbAAb cbAb cbAbcbAAb`
	`bAbcbAAbc bAbc bAbcbAAbc`
	`AbcbAAbcb Abcb AbcbAAbcb`
	`bcbAAbcbA bcbA bcbAAbcbA`
	`cbAAbcbAb cbAb cbAAbcbAb`
	`bAAbcbAbc bAbc bAAbcbAbc`
	`AAbcbAbcb Abcb AAbcbAbcb`

lattice:

                     R3 -- D3 -- G3 -- N3 -- m3
                      |     |     |     |     |
   r1 -- d1 -- g1 -- n1 -- Ma -- Sa -- Pa -- R4 -- D4 -- G4 -- N4 -- m4
                           |      |     |     |     |
                           r2 -- d2 -- g2 -- n2 -- m2
5
|
1 -- 3
                     10     5     5    15    45
                      / --- / --- / --- / --- /
                      9     3     4     8    32
                      |     |     |     |     |
   256   128   32    16     4     1     3     9    27    81    243   729
    / --- / --- / --- / --- / --- / --- / --- / --- / --- / --- / --- /
   243    81   27     9     3     1     2     8    16    64    128   512
                            |     |     |     |     |
                           16     8     6     9    27
                            / --- / --- / --- / --- /
                           15     5     5     5    20

This scale could be considered a detempering of the following OTC scales with two step sizes:

OTC 17L+5s (A=L, b=L, c=s)
superpyth MOS
AbcbAbcbAbcbAAbcbAbcbA
LLsLLLsLLLsLLLLsLLLsLL

OTC 15L+7s (A=s, b=L, c=L)
porcupine MODMOS
AbcbAbcbAbcbAAbcbAbcbA
sLLLsLLLsLLLssLLLsLLLs

OTC 12L+10s (A=L, b=s, c=L)
pajara MODMOS
AbcbAbcbAbcbAAbcbAbcbA
LsLsLsLsLsLsLLsLsLsLsL (form 1)